I've just finished migrating my website from ASP.NET to ASP.NET Core. This choice was mostly driven by my personal curiosity, but in fact, .NET Core is going to be a big thing for the development future. Also, if you're a .NET developer, I believe this is going to be a mandatory path sooner or later.
As a matter of fact, Microsoft is doing lots of good moves (IMHO) in order to constantly compete in a more and more polyhedric development world which sees newborn languages and technologies everyday. Starting from acquiring Xamarin, then GitHub, and now consolidating the .NET Core framework.
I started with classic Web Forms applications, then moved to MVC while winking to Web Apis, and then ended up with ASP.NET Core. My first host was ARUBA: an italian provider which offers good services at very competitive prices. Unfortunately ARUBA does not support .NET Core yet ('till today at least) so I decided to move to ASPNIX.
ASPNIX is a bit more expensive but offers all the support for .NET Core applications and a couple of deployment tools as well. The only negative note is that if you want an SSL certificate you need a dedicated IP address, but you can notice this by digging inside the HOWTOS/FAQ pages only because it's not very clear from the offer page. And guess what? Dedicated IP addresses costs extra money that you have to sum to the domain activation (or transfer in my case) and hosting fee. So basically you pay for the SSL certificate, but this is only my point of view. I've choosen a monthly fee, so depending on the QoS I'll decide if I stay or not.
By the way, .NET Core is awesome. I'll write a dedicated article later about all the pros and cons of using/migrating to it. For the moment, the thing I like the most is the built in Dependency Injection system (will miss you Zenject and Ninject).